I don't believe that it is said at any point that they army approaching Caemlyn is definitely coming to lay siege. Sure, Elayne orders that preparations just in case, but that's just being sensible. There's nothing to say that whoever it is isn't coming with the intention of helping her.

That said, out of all the possibilities that people have already named, it seemed to me at least like the most probable answers are either the loyal Andorans sent east by Rahvin, Gawyn, or even better, a combination of the two. Gawyn is well aware by the end of book 6 that Elaida would prefer him dead, and word that Elayne has declared for the throne might be what it would take to pry him from the Tower. He would surely bring his Younglings with him and even likely increase their numbers throughout the march. Although they would be unlikely to have reached 20,000 by the time they reached Andor, I think it's possible if they met up with the Andoran loyalists to the east, who would certainly follow Gawyn.

I think it unlikely that the Andoran Houses could summon an army that large, especially in the dead of winter. Remembering how awestruck people are of national armies hitting the five digit mark in the earlier books, that only SOME houses could raise a force of that size seems unlikely. The Whitecloaks would have to come from the west or south as they do not have access to Travelling. As was pointed out earlier, the Shaido have numerous foes nearby that could crush them with ease. The Asha'man have no need for 20,000 men. Egwene is outside Tar Valon. The Murandians are busy playing the Game of Houses with the Band of the Red Hand, which Talmanes said he is moving southward. The other nations are too busy with their own problems.